Debye-Hiickel screened Coulomb potential

Figure 13. The screened Coulomb potentials [11c] The Naive Approximation (full curve) is compared with the Debye-Hiickel potential (dashed curve), both at c=50mM. The unscreened potential (wide full curve) is also shown for comparison.

To investigate the effect of the Debye-Huckel approximation on the solution properties, Stevens and Kremer [152] performed molecular dynamics simulations of salt-free solutions of bead-spring polyelectrolyte chains in which the presence of counterions was treated via a screened Coulomb potential, and compared the results with their simulations with explicit counterions [146,148]. To elucidate the effect of the Debye-Hiickel approximation, the dependence of the mean square end-to-end distance, R ), osmotic pressure, and chain structure factor on polymer concentration was examined. Stevens and Kremer found that (i ) tends to be larger at low densities for DH simulations and is smaller at higher densities. However, the difference in (i ) between DH simulations and simulations with explicit counterions is within 10%. This trend seems to be a generic feature for all N in their simulations. The functional form and density dependence of the chain structure factor are very close in both simulations. The most severe Debye-Huckel approximation affects the dependence of the osmotic pressure on polymer concentration. It appears that in the DH simulations not only is the magnitude of the osmotic pressure incorrect, but also the concentration dependence is wrong. 

The electric potential given by Equation 3.33 is called the extended Debye-Htickel potential and is plotted in Figure 3.3b. For comparison, the result of Equation 3.18 is included in the figure. The finite size of the ion modifies the prefactor by a factor of exp(iefl)/(1+ku), and the beginning of the decay of the potential is shifted to the ion diameter. The distance dependence is the same as the screened Coulomb potential for r > a. For a 0, Equation 3.33 reduces to the Debye-Hiickel potential. 

Alternatively, there has been a revival of Debye-Hiickel (DH) theory [196-199] which provides an expression for the free energy of the RPM based on macroscopic electrostatics. Ions j are assumed to be distributed around a central ion i according to the Boltzmann factor exp(—/ , - y.(r)), where y(r) is the mean local electrostatic potential at ion j. By linearization of the resulting Poisson-Boltzmann (PB) equation, one finds the Coulombic interaction to be screened by the well-known DH screening factor exp(—r0r). The ion-ion contribution to the excess free energy then reads... 

As shown by Debye and Hiickel, due to the strong electrostatic interaction between the ions in a solution, the positions of the ions are correlated in such a way that a counterion atmosphere appears aronnd each ion, thns screening its Coulomb potential. The energy of formation of the counterion atmospheres contribntes to the free energy of the system called correlation energy. ... 

The electrostatic potential energy between two ions given by Equations 3.19 and 3.20 is called the Debye-Hiickel potential energy. The collective effect of the ions in the solution is to screen the Coulomb interaction between a pair of ions given by Equation 3.5 resulting in the screened electrostatic interaction given by Equation 3.19. 

So far, very dilute solutions have been considered such that the interaction between ions is only coulombic. When other (unreactive) ions are nearby, the direct interaction between ionic reactants is partially screened and was first developed by Debye and Hiickel [91]. They showed that the potential energy, eqn. (39), is modified and becomes... 

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